c 2004 Cambridge University Press
J. Fluid Mech. (2004), vol. 505, pp. 235–248. DOI: 10.1017/S0022112004008407 Printed in the United Kingdom
235
The leading effect of fluid inertia on the motion
of rigid bodies at low Reynolds number
By A. M. L E S H A N S K Y1 , O. M. L A V R E N T E V A2
1
AND
A. N I R2
Division of Chemistry & Chemical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
2
Department of Chemical Engineering, Technion – IIT, Haifa 32000, Israel
(Received 18 January 2002 and in revised form 11 December 2003)
We investigate the influence of fluid inertia on the motion of a finite assemblage of
solid spherical particles in slowly changing uniform flow at small Reynolds number,
Re, and moderate Strouhal number, Sl. We show that the first effect of fluid
√ inertia on
particle velocities for times much larger than the viscous time scales as Sl Re given
that the Stokeslet associated with the disturbance flow field changes with time. Our
theory predicts that the correction to the particle motion from that predicted by the
zero-Re theory has the form of a Basset integral. As a particular example, we calculate
the Basset integral for the case of two unequal particles approaching (receding) with
a constant velocity along the line of their centres. On the other hand, when the
Stokeslet strength is independent of time, the first effect of fluid inertia reduces to a
higher order of magnitude and scales as Re. This condition is fulfilled, for example,
in the classical problem of sedimentation of particles in a constant gravity field.
1. Introduction
The understanding and modelling of the complex flows of dispersed material is
paramount in the design and process control of various technological applications,
such as fluidization, sedimentation, slurry transportation etc. While the transient or
oscillatory motion of a single particle has attracted wide attention through the years
(Tchen 1947; Mei & Adrian 1992; Lovalenti & Brady 1993; Chang & Maxey 1994;
Coimbra & Rangel 2001), the theory concerning the analogous behaviour of multiparticle systems is still quite limited, probably due to analytical complexity arising
from many-body interactions and the time-dependence of the flow domain geometry.
Machu et al. (2001) considered the behaviour of single drops and pairs of drops made
from dilute suspensions of microscopic particles with negligible effects of inertia
and found a remarkable analogy with the motion of drops of homogeneous liquids.
Leichtberg et al. (1976) developed a strong-interaction theory to describe the coaxial
gravitational settling of three rigid spheres at low Reynolds numbers, Re. It was
suggested that the unsteady Basset history-type contribution to the Stokes drag force
in the long-time limit results from the slow temporal change of the domain geometry
due to inter-particle interactions. Although they claim that a precise treatment of this
force for a multi-particle problem would need to take into account the instantaneous
boundaries of the other particles, the classical single-sphere Basset term (Landau &
Lifshitz 1988) was retained in the equation of motion for each particle. Their results
show the deviation from the zero-Re theory grows as t 1/2 due to the cumulative effect
of the Basset force. Feng & Joseph (1995) studied the unsteady motion of solid bodies
236
A. M. Leshansky, O. M. Lavrenteva and A. Nir
in more complex flows. They argued that in cases where the unsteadiness is caused
by hydrodynamic inter-particle or wall–particle interactions, the unsteady inertial
terms, ∂v/∂t dominate over the convective ones, v · ∇v. By means of direct numerical
simulations of the transient Stokes equations they showed that the unsteady forces
may completely change the dynamics of the particles’ motion in some cases. Although
they mention that if the entire fluid inertia is considered the solution may look quite
different from what they have obtained taking into account the unsteady fluid inertia
only, the question of when the nonlinear convective inertia terms should be retained
in the governing equations remained open.
Hinch & Nitsche (1993) showed that mutual interactions between fluctuating
colloidal particles may give rise to a nonlinear drift force when the fluid inertia is not
entirely neglected. Moreover, they found that this force, when integrated over all the
spectrum of frequencies, results in a net drift interaction on ensemble average. The
nonlinearities due to time-dependence of the flow domain and the convection within
the bulk fluid were neglected by considering oscillations with small amplitudes.
Recently, Pieńkowska (2001) studied the structure of an unsteady flow generated
in a viscous fluid by a finite number of solid spheres translating in it with prescribed
velocities at small Re. A multiple scattering approach was used to describe the effects
of mutual non-stationary interactions in terms of velocity tensors and induced forces
distributed on the sphere surfaces. The leading-order effect of the unsteady inertia
on the forces induced on the particles was found. The fully linearized problem was
treated in both this work and Hinch & Nitsche (1993), i.e. the relative positions of
the particles were assumed to be independent of time during the time interval under
consideration. Thus, their results are applicable to small time intervals and to the case
of small-amplitude particle oscillation.
In the most recent study Coimbra et al. (2004) investigated experimentally history
effects in particle response to a high-frequency small-amplitude oscillatory flow. The
theoretically determined behaviour of the Basset drag on a single particle freely
moving in a unform time-dependent background flow from the earlier studies has been
unequivocally validated in these experiments. Their results also show that in the case of
the high-frequency background flow the hydrodynamic interaction between particles
is negligible even for moderate separation distances of the order of particle size.
This paper presents a preliminary attempt to analyse the leading effect of fluid
inertia on the motion of a finite assemblage of rigid spherical particles in a uniform
time-dependent flow at low Reynolds numbers during the time interval when the
mutual positions of the particles may significantly change. Our focus is on the case of
a moderate number of particles in the swarm when the change in the mutual positions
may have a substantial effect on the dynamics. The study is performed by means
of singular perturbation methods based on the well-established theory of matched
asymptotic expansions. It is predicted that as a result of unsteadiness
√ there might
be an inertial contribution to the particles’ velocities proportional to Sl Re, where
Sl is the Strouhal number, given that the net Stokes drag force exerted on them by
the fluid varies considerably with time. Some remarks concerning the influence of the
fluid inertia on motion of particles suspended in more general time-dependent flows,
such as linear and parabolic flows are also included in the discussion.
2. Statement of the problem
Consider the evolution of a swarm of n rigid particles of mass mi in a spatially
uniform time-dependent unbounded flow of a viscous Newtonian fluid with density
The leading effect of fluid inertia on the motion of rigid particles
237
and the viscosity equal to ρ and µ, respectively. Let x = (x1 , x2 , x3 ) be a radius vector
to a point in the laboratory coordinate system with some chosen origin. If we suppose
that the particles are spherical, then the domain occupied by the continuous phase
may be represented by D(t) : {x ∈ 3 : |x − Z i (t)| > ai , i = 1, 2, . . . , n}, where Z i (t)
denotes a radius vector to the center of the ith particle and ai stands for the particle
radius. The size of the swarm can be characterized by L(t) = maxi,j |Z i (t) − Z j (t)|.
The velocity field v(x, t) = (v1 , v2 , v3 ) and the pressure p(x, t) are described by the
following equations and boundary conditions:
∂v/∂t + v · ∇v = −ρ −1 ∇p + νv + g,
x ∈ D(t),
(1)
∇ · v = 0,
x ∈ D(t),
(2)
v = ui (x, t),
x ∈ Γi (t),
(3)
v → v ∞ (t),
p → p ∞ (x, t)
v(x, 0) = 0,
as
|x − X(t)| → ∞,
x ∈ D(0),
(4)
(5)
where Γi (t) denotes the surface of the ith particle, ui (x, t) is the velocity at points
on the surface of ith particle, and X(t) is the position vector for the centre of mass
of the swarm. We also assume that {p∞ (x, t), v ∞ (t)} is a solution of (1)–(2) in 3
with v ∞ (0) = 0. Since the particles are rigid, ui = U i (t) + Ω i (t) × r i , where U i and Ω i
are the linear and angular velocities of motion and r i = x − Z i (t) is the radius vector
with its origin at the centre of the ith particle. We also assume that the body force
is spatially homogeneous and time-dependent with density g(t). The instantaneous
force, F i , and torque, T i , exerted on the ith particle by the flow are given by
σ · n ds,
(6)
Fi =
Γi
r i × (σ · n) ds,
Ti =
(7)
Γi
where σ = −pI + µ(∇v + (∇v)T ) is the stress tensor. The evolution of the particle
swarm is governed by following kinematic conditions:
U i = Ż i (t),
i = 1, 2, . . . , n,
(8)
where the over-dot stands for the time derivative.
There are two classical problems associated with the equations outlined above:
(i) The resistance problem: to determine forces and torques exerted by the flow on
each particle for a given set of particle velocities, {U i (t), Ω i (t)}.
(ii) The mobility problem: to determine U i (t) and Ω i (t) of particles that are freely
suspended in the ambient fluid, for prescribed external forces and torques.
For problem (ii) the dynamic force and torque balances must also be specified,
dU i
= F i + mi g,
(9)
dt
dΩ i
= Ti,
Ii
(10)
dt
where Ii denotes the moment of inertia of particle i. To complete the formulation of
problem (ii), the initial data for the positions and velocities of the particles should be
provided:
mi
Z i (0) = Z i0 , U i (0) = 0, Ω i (0) = 0,
i = 1, 2, . . . , n.
(11)
238
A. M. Leshansky, O. M. Lavrenteva and A. Nir
In this paper we shall address both problems formulated above with a special attention
given to the mobility problem (ii).
There are three different timescales associated with the problem formulated above
that are important for our analysis: the viscous relaxation time on the lengthscale
associated with the swarm, tν = L2 /ν; the Stokes time associated with the motion
of a single particle under the influence of a body force, tS = ν/ag(λ − 1), where
g = max|g(t)|, and λ = ρs /ρ is a typical density ratio; and the time related to the
temporal change in the external forces or uniform flow, text (period in the periodic
case). Previous theoretical studies of the multi-particle problem (Hinch & Nitsche
1993; Pieńkowska 2001) were focused on the case when the particles’ mutual positions
may be regarded as independent of time. The first influence of inertia in this case is
described by the fully linearized unsteady Stokes equations. In contrast to this, our
focus is on the larger timescales, when the evolution of the geometry of the swarm
cannot be neglected.
The following generalized scaling is chosen: the radius of the reference particle, a,
for length; the characteristic velocity associated with either the particle motion or
applied body forces, u, for velocity; u/a for angular velocity; and µ
u/a for pressure.
The characteristic time τ is associated either with the slow change (to compare with
a viscous time) of the flow due to the motion of particle, τ = tS , or with the variation
of the external force, τ = text 6 O(tS ). Assume further that tν τ , i.e. that the
quasi-steady velocity field in the vicinity of the swarm is immediately established.
In dimensionless form the undisturbed velocity and pressure fields should satisfy
∂v ∞
(12)
= −∇p ∞ (x, t) + η(t), x ∈ 3 ,
Re Sl
∂t
where Re = ua/ν is a Reynolds number, Sl = (a/
u)/τ is a Strouhal number and
η = g/(ν u/a 2 ) stands for the dimensionless density of the body force. Note that if the
external force and flow do not change with time and the unsteadiness of the problem
is caused solely by the evolution of the geometry, then Sl = 1.
Following Lovalenti & Brady (1993), we pose the problem formulation in a comoving coordinate frame with its origin at the instantaneous centre of a reference
particle belonging to the swarm, i.e.
t
U(τ ) dτ,
(13)
x = x −
0
where U denotes the translational velocity of the reference particle. Since we want to
focus on a velocity field that vanishes far from the swarm, we consider the disturbance
velocity and pressure fields and introduce the new variables
v = v − v∞,
p = p − p∞ .
(14)
Thus, after omitting the primes, the problem formulated for the disturbance field in
dimensionless form is
Re(Sl∂v/∂t + v · ∇v + f (v, v ∞ )) = −∇p + v,
x ∈ D(t),
(15)
∇ · v = 0,
x ∈ D(t),
(16)
v = ui − v ∞ ,
x ∈ Γi (t),
(17)
v, p → 0
where
f (v, v ∞ ) = (v ∞ − U) · ∇v.
as
|x| → ∞,
(18)
The leading effect of fluid inertia on the motion of rigid particles
239
Kinematic conditions (8) and initial conditions (11) in dimensionless form become
Ż i (t) = Sl−1 U i ,
Z i (0) = Z i0 , U i (0) = 0, Ω i (0) = 0,
v(x, 0) = 0,
i = 1, 2, . . . , n,
(19)
i = 1, 2, . . . , n,
(20)
x ∈ D(0),
(21)
while the force and torque balances (9)–(10) combined with (6)–(7) become
dU i
i η(t),
i Re Sl
= (σ + σ ∞ ) · n ds + m
m
dt
Γi
dΩ i
r i × [(σ + σ ∞ ) · n] ds,
=
Ii Re Sl
dt
Γi
(22)
(23)
i is the mass of particle i non-dimensionalized by the product ρa 3 , Ii is the
where m
moment of inertia of the particle scaled by the product ρa 5 and σ ∞ is the stress
tensor corresponding to v ∞ and p ∞ .
3. Construction of the solution for small Re
3.1. Zero-order approximation
In this paper we shall consider the solution generated under the following conditions:
a2
, Sl Re.
(24)
L2
The first condition is equivalent to the condition tν τ mentioned above while
the second one means that the transient inertia term prevails over the convective
inertia in (15). A similar limitation has been already considered in a number of works
(Leichtberg et el. 1976; Lovalenti & Brady 1993), corresponding to an isolated particle
case, as well as in the multi-particle problem (Pieńkowska 2001). Another restriction
concerns the size of the assemblage, which should remain relatively compact over
durations of interest, i.e. L(t) ∼ O(L(0)) a(Sl Re)−1/2 . Although, there is no general
criterion that would guarantee that inter-particle spacing remains bounded, there is,
nevertheless, a number of well-studied cases, where a collection of particles remains
compact for times far exceeding the typical viscous time. For example, the existence
of the compact triplet in the 3-particle gravitational settling in a vertical plane for
an extremely wide range of initial configurations was shown to be a consequence
of a transient chaotic saddle in the phase space (Jánosi et al. 1997). Some studies
concern periodic and quasi-periodic dynamics of 3-particle sedimentation (Hocking
1964; Caflisch et al. 1988; Golubitski, Krupa & Lim 1991; Snook, Briggs & Smith
1997), while Durlofsky, Brady & Bossis (1987) constructed periodic solutions for
configurations of four and eight particles that can be anticipated solely from the
symmetry of Stokes equations.
The zero-order approximation is the quasi-steady solution of (15)–(23) under the
assumption of Re = 0. In this case, the dependence of forces (6) and torques (7) on the
particle positions and velocities is local in time, while the dependence on the velocities
is linear. Thus, when the solution of problem (i) is known, the particle velocities can
be found by solving a system of linear algebraic equations. The evolution of the
assemblage configuration can be obtained by solving a system of ordinary differential
equations (8). This is the standard procedure to solve the mobility problem under the
quasi-stationary approximation, and, in principle, it can also be employed for studying
Sl Re 240
A. M. Leshansky, O. M. Lavrenteva and A. Nir
nonlinear cases. However, for non-zero Re the dependence of forces and torques on
velocities is not linear and, moreover, it is not local, i.e. the force acting on a particle
depends not only on instantaneous positions and velocities but also on the history of
the process. This makes the inversion of the resistance problem a rather complicated
task. In the present paper we consider the mobility problem directly without solving
an auxiliary resistance problem.
Under the approximation of quasi-stationary Stokes flow, the positions of the
particles Z i vary with time; although the velocity field is found as a solution of
a non-evolutionary problem, it depend on time parametrically via the evolution
of the problem domain. Of course, the quasi-steady solution cannot satisfy the
initial condition (20)–(21) and a smaller timescale of O(tν ) exists, representing the
initial period of rapid acceleration from rest. Re-scaling of the dimensionless time
as t ∗ = (Sl Re)−1 t shows that in the initial period all three terms in (22) are of the
same order of magnitude, the unsteady term in (15) is O(1), while the nonlinear
convective term is of O(Re) and can be neglected. Therefore during this initial period
the unsteady inertial, virtual-mass and the Basset forces would be equally operative
(Leichtberg et al. 1976). The right-hand side of (19) becomes of O(Re) as well hence
the change in particle positions can be neglected. Thus, the initial transient period
can be described by the fully linearized problem studied by Leichtberg et al. (1976)
and by Lovalenti & Brady (1993), corresponding to an isolated particle, as well as in
the multi-particle problem (Pieńkowska 2001).
In what follows we shall focus on the long-time behaviour of the assemblage, t tν ,
when the quasi-steady velocity and pressure fields in the vicinity of the swarm have
already developed. Therefore, we neglect the initial short transient motion and we
do not attempt to satisfy the initial conditions exactly everywhere. More specifically,
we are interested in the leading-order inertial corrections to the quasi-steady particle
velocities ui = u0i + u1i + o(), (Sl, Re) 1.
It follows from (12) that the undisturbed pressure can be readily found for an
arbitrary Re as
p∞ (x, t) = p0∞ (t) + η · x − Re Slv̇ ∞ · x,
(25)
p0∞ (t)
is an arbitrary time-dependent uniform pressure. For Re = 0 the problem
where
formulated in (15)–(23) is a Stokes problem. Using the boundary integral representation of the velocity field, v 0 , corresponding to Re = 0, and eliminating the double layer
integral (Pozrikidis 1992), we arrive at
0
1
0
(26)
σ ( y) · n + σ ∞
v =−
0 ( y) · n · G(x − y) ds( y),
8π S
where σ ∞
solution of a homogeneous Stokes
0 denotes a stress corresponding to the equation with the undisturbed velocity v ∞ , S = i Γi is a multiply-connected solid
surface and G is the Oseen–Burgers tensor or Stokeslet with components
Gij (x) =
δij
xi xj
+ 3 .
r
r
Note that (26) is valid for an arbitrary Stokesian velocity field v ∞ . For the special case
∞
of a spatially uniform flow, σ ∞
0 · n is just a constant pressure, −p0 n, of no dynamic
significance and does not contribute to the the integral in (26) since G is solenoidal.
Far from the swarm, |x| | y|, the leading term of the disturbance field can be
found as in Kim & Karrila (1991), by using the multi-pole expansion of the integral
241
The leading effect of fluid inertia on the motion of rigid particles
representation for the velocity and pressure fields,
F 0 · G(x)
+ O(|x|−2 ),
8π
F 0 · P(x)
p0 = −
+ O(|x|−3 ),
8π
v0 = −
(27)
(28)
where P is the pressure vector with components Pi = 2xi /r 3 and
F 0 = σ 0 · n ds
(29)
S
is a net Stokes drag force exerted by the fluid on the particles.
The leading term of the disturbance in (27) represents the velocity field induced by
a Stokeslet of strength F 0 at r = 0 and it is present if particles and fluid exert a net
force on each other. Using (22) and (12) it follows that at vanishing Re
i.
(σ 0 + σ ∞ ) · n ds =
(σ 0 − (η · x)I) · n ds + O(Re) = −η(t)
m
i
Γi
Γi
i
i
Therefore, taking into account that
i λ−1
(η · x)n ds = m
i η,
Γi
where λi denotes the dimensionless density of the ith particle non-dimensionalized by
ρ, we finally arrive at
0
i 1 − λ−1
σ 0 ( y) · n ds( y) = −η(t)
m
F =
.
(30)
i
i
Γi
i
Note that all the identities above are valid for any Stokes flow irrespective of what
boundary conditions are specified on the particle boundary. Thus, they are equally
applicable to both problems (i) and (ii). For the mobility problem (ii), obviously the
disturbance flow is force free if there are no body forces (η = 0) or when the particles
are neutrally buoyant (λi = 1).
Consider now the case of small but non-zero Re. Our goal is to construct the leadingorder correction to the quasi-stationary solution outlined above under limitations (24).
It is obvious that a regular expansion of the velocity field in integer powers of Re
breaks down since ∂v 0 /∂t decays as r −1 far from the assemblage and thus the first
correction of the velocity field, v 1 , does not possess solutions that are bounded
at infinity. Thus, a singular perturbation method will be employed to construct a
uniformly valid asymptotic expansion of the solution. Following a well-established
procedure (van Dyke 1975) we first construct an outer expansion of the velocity and
pressure fields.
3.2. The first term of the outer expansion
Let ξ = εx, V (ξ , t) = v(ε−1 ξ , t), P (ξ , t) = p(ε−1 ξ , t), where 0 < ε(Sl, Re) 1 is some
small parameter. In terms of the outer spatial variables equation (15) becomes
∂V
(31)
+ εV · ∇ξ V + ε f (V , V ∞ ) = −ε∇ξ P + ε2 ξ V .
Re Sl
∂t
242
A. M. Leshansky, O. M. Lavrenteva and A. Nir
We assume that the outer velocity and pressure fields have the following asymptotic
expansions:
V = f0 (ε)V 0 + f1 (ε)V 1 + . . . ,
(32)
P = ε(f0 (ε)P 0 + f1 (ε)P 1 + . . .),
where limε→0 fn+1 /fn √= 0 and the functions V n and P n vanish as |ξ | → ∞. Then,
with the choice ε = Sl Re the first term of the outer expansion should satisfy the
transient Stokes equation
∂V0
(33)
= −∇ξ P 0 + ξ V 0 .
∂t
f0 (ε) can be readily obtained from the matching condition, requiring that
F 0 · G(x)
,
|ξ |→0
|x|→∞
8π
F 0 · P(x)
.
lim εf0 (ε)P 0 (εx, t) = lim p 0 = −
|ξ |→0
|x|→∞
8π
lim f0 (ε)V 0 (εx, t) = lim v 0 = −
(34)
(35)
Thus, it follows from the above matching conditions that f0 = ε.
0 (ξ , s) and P0 (ξ , s) be the Laplace images of the first term of the outer
Let V
velocity and pressure fields, respectively, where s is a Laplace variable. It can be
readily verified that the fundamental solution of the transient Stokes equation with
the singularity as ξ → 0 corresponding to the varying point force (Kim & Karrila
1991), given by
0 0 (ξ , s) = − F · G(ξ ; s) ,
V
8π
; s)
0 · P(ξ
0
F
P (ξ , s) = −
,
8π
; s)
satisfies the above equation and the matching conditions (34)–(35). Here G(ξ
denotes a transient Oseen tensor with components
Gij (ξ ; s) =
√ −√s ξi ξj
4
[1
−
(1
+
s)e
] 2
3 s 2
√
√
2
ξi ξj
2
− s
+ 3 2 [(1 + s + s)e
− 1] δij − 2 ,
s
(36)
and
i (ξ ; s) = Pi (ξ ) = 2 ξi ,
P
3
where = εr = |ξ |.
Expanding εv 0 near = 0, then re-writing it in terms of the inner spatial variables
up to O(ε) and transforming back to the time domain, t, we find the outer(inner)
limit of the inner(outer) solution as
t
Ḟ 0 (ζ )
4
1
F 0 · G(x) − ε √
dζ
+ o(ε),
(37)
V =−
8π
3 π −∞ (t − ζ )1/2
where we assumed that F 0 (−∞) = 0, which indicates that we neglect the influence of
the short-lived initial transient on the long-time behaviour. The second term of O(ε)
is a generalized Basset inertial term, since it accounts for all inter-particle interactions.
For instance, we will show in § 3.4 that this term has a non-zero value in the case of a
steady translation of particles towards each other, which is a case when the classical
The leading effect of fluid inertia on the motion of rigid particles
243
Basset contribution is zero. It can be readily shown that in the case of a single particle
moving due to prescribed forcing F 0 (t) this expression will give rise to a classical
Basset integral (Kim & Karrila 1991).
The form of (37) and (32) suggests that the second term of the inner expansion
should be of the order of ε = (Sl Re)1/2 ,
(v, p) = (v 0 , p 0 ) + ε(v 1 , p 1 ) + . . . .
A similar expansion is expected for the variables U i and Ω i .
3.3. The second term of the inner expansion
For the mobility problem (ii), the second term in the inner expansion of the velocity
and pressure fields should satisfy the Stokes equation and the following conditions:
−∇p 1 + v 1 = 0,
x ∈ D,
(38)
∇ · v = 0,
x ∈ D,
(39)
x ∈ Γi ,
(40)
1
v =
1
u1i (t),
−1
v → (8π) b(t)
1
where
4
b(t) = √
3 π
t
−∞
at
|x| → ∞,
Ḟ 0 (ζ )
dζ,
(t − ζ )1/2
(41)
(42)
and where the boundary conditions (40) are applied at the disturbed particle surfaces,
Γi (t), corresponding to their perturbed positions, Z i (t) = Z 0i (t) + ε Z 1i (t). Given that
i U̇ i | ∼ |Ii Ω̇ i | =
particle inertia contributes the higher-order corrections, i.e. when |m
O(1), the dynamic force and torque balances (22)–(23) at O(ε) give
σ 1 · n ds = 0,
(43)
Γi
r i × (σ 1 · n) ds = 0.
(44)
Γi
Thus, the mobility problem (ii) formulated in (38)–(41) has a simple solution v 1 =
u1i = (8π)−1 b(t), p 1 = Const, which satisfies (43)–(44) on any Γi (t). Thus, the firstorder effect of fluid inertia on the motion of freely suspended particles results in
an additional translational velocity, which is the same for all particles, regardless
of the instantaneous configuration and velocities. Note that in the classical case of
gravitational settling of particles in a constant gravity field, given that the assemblage
remains compact during its fall, L ε−1 , it follows from the conclusions of § 3.1 that
F 0 is a constant vector and, therefore, b(t) = 0 in (41). Thus, the problem (38)–(43)
has only a trivial solution with u1 = 0 and it is anticipated that, in this particular
case, the first effect of the fluid inertia is of the higher order of O(ε2 ).
When a swarm of non-neutrally buoyant particles, λi = 1, is submerged into a
harmonically oscillating uniform background flow, η(t) ∼ eiωt , with low oscillation
frequency, Re = ωa 2 /ν 1,
√ the result will be a periodic correction to the particle
velocities, u1 ∼ ε(1 − i)eiωt / ω in the long-time limit. Although there is a phase lag
in the inertia-induced particle response, it does not contribute to the departure in
particle spacings, Z 1i (t), from those predicted by the zero-Re theory at O(ε).
244
A. M. Leshansky, O. M. Lavrenteva and A. Nir
3.4. Motion with prescribed velocities
In the previous section we have shown that the temporal change in the domain
geometry does not always perturb the quasi-steady zero-Re solution at O(ε). An
assemblage of freely suspended particles does not experience a Basset force at this
order and the inertia-induced correction to their velocities is of o(ε). On the other
hand, for the resistance problem (i), when the particle velocities, ui (t), are prescribed
the aforementioned Basset force may be operative. To demonstrate this idea, consider
the time-dependent motion of a swarm in a fluid which is quiescent at infinity,
v ∞ , p∞ = 0, in the absence of body forces, η = 0. The hydrodynamic force F i exerted
by the fluid on the ith particle is given by
σ · n ds + O(Re Sl),
(45)
Fi =
Γi
where the correction corresponds to the magnitude of the particle inertia.
The zero-order approximation of this problem is the solution of (15)–(21) for a
given set of values, ui (t). The first term in the outer expansion has exactly the same
form as in (37) while the second term of the inner expansion of the velocity and
pressure fields should satisfy (38)–(41) with u1i = 0, where b(t) is given by (42), while
the boundary conditions are applied at the particle surfaces, Γi0 (t) corresponding to
their known positions, Z i (t) = Z 0i (t). The O(ε) correction to the zero-Re drag force
acting on ith particle is
σ 1 · n ds.
F 1i =
Γi0
F 1i
It is readily seen that
is identical to the Stokes drag force exerted on the ith
particle by the uniform time-dependent flow, that equals (8π)−1 b(t) at infinity, past a
fixed assemblage of particles.
As an example, let us calculate the generalized Basset force arising in the co-axial
steady approach of two particles, ui (t) = Ui e, where Ui are constants. In the Stokes
approximation the resistance problem corresponding to the uniform co-axial flow
past two separated equal spheres was first solved exactly in bi-spherical coordinates
by Stimson & Jeffery (1926). Using bi-spherical coordinates, the axial components,
Fi0 , Fi1 and b, can be found as functions of the dimensionless separation distance,
Z(t) = Z1 (t) − Z2 (t) by the same technique. Since the solution at leading order is
quasi-stationary, i.e. it depends on time via the evolution of the problem domain and
the Basset inertial term, (42) can be written as
t
t
4
dF 0 (Z(τ ))
Ż
Ḟ 0 (τ )
4
√
dτ
=
dτ,
(46)
b(t) = √
1/2
dZ
(t − τ )1/2
3 π −∞ (t − τ )
3 π −∞
where F 0 = 2i=1 Fi0 .
In figure 1 we present the results of calculations for b(t) for two spheres with
a radius ratio R = a2 /a1 = 3.5 for four different modes: (a) the smaller particle
approaches the fixed larger one with the velocity U1 = 1, Z0 = 30; (b) the smaller
particle moves away from the fixed larger particle with velocity U1 = 1, Z0 = 0.1; (c)
the larger particle approaches the fixed smaller one with U2 = 1, Z0 = 100; (d) the
larger particle moves away from the fixed smaller particle with U2 = 1, Z0 = 1. Note
that in case (a) F10 and b(t) have different signs and thus fluid inertia reduces the
drag force which the fluid exerts on the moving smaller particle, while the drag force
on the stationary larger particle increases. In case (b) the situation is opposite: F10
245
The leading effect of fluid inertia on the motion of rigid particles
(a)
(b)
3
1.5
2
1.0
b(t)
1
0.5
0
0
0
5
10
15
20
25
0
30
5
10
(c)
15
20
25
30
(d )
0.2
0.4
0.2
0
0
–0.2
–0.2
–0.4
b(t)
40
60
80
t
100
0
10
20
30
t
40
50
60
Figure 1. The Basset inertial term b(t) in (46) vs. time for two unequal particles with a radius
ratio R = a2 /a1 = 3.5. (a) The smaller particle approaches the larger stationary one with
a velocity U1 = 1 and with initial separation Z0 = 30; (b) the small particle moves away
from the stationary larger one with a velocity U1 = −1 and with initial separation Z0 = 0.1;
(c) the larger particle moves towards the smaller particle with a velocity U2 = −1 and the
initial separation is Z0 = 100; (d) the larger particle moves away from the small particle with
a velocity U2 = 1 and where the initial separation is Z0 = 1.
and b(t) have the same signs and thus fluid inertia results in an increase of the drag
force on the moving smaller particle, while the drag force on the stationary larger
particle reduces. Although in cases (c) and (d) b(t) changes sign when the particles
are close enough, due to non-monotonic dependence of the Stokeslet strength, F 0 ,
on separation distance, the effect of fluid inertia on the drag force at large spacing
is qualitatively the same as in (a) and (b), respectively. Recall that for the resistance
problem for two equal particles, the four modes discussed above for the unequal
particles reduce to only two modes due to symmetry: a particle moves towards or
away from the neighbouring fixed particle.
Analysis of a similar example (the co-axial translation of three particles) under the
limitation that the mutual positions of the particles do not change significantly
√ with
time (Pieńkowska 2001), also shows the existence of the Basset force at O( Sl Re)
due to unsteadiness caused by inter-particle interactions.
4. Discussion
In the present paper we investigate the influence of fluid inertia on the dynamics
of a finite collection of rigid spherical particles submerged in an arbitrary slowly
changing uniform viscous flow in the long-time limit. Our focus is on the case of a
swarm of a moderate number of particles with inter-particle separations comparable
to their dimensions, when the change in mutual positions may have a substantial
effect on the dynamics. For this case, the velocities of the particles in the swarm are
246
A. M. Leshansky, O. M. Lavrenteva and A. Nir
comparable to the velocity of an isolated particle, which we choose for a velocity scale,
u. On the other hand our considerations are restricted to the time interval during
which the swarm remains compact, i.e. the dimension of the swarm is comparable to
its initial value. Since the rate of change of inter-particle separations may be either
of the same order of magnitude as the individual velocities or much less, but never
much greater, this time interval is > O(τ ), τ = a/
u.
In the case Sl Re a 2 /L2 , Sl Re, we found that the flow field possesses a tworegion structure. In the vicinity of the assemblage there exists, to a first approximation,
a quasi-steady flow field region that we term the inner√
region and which corresponds
to inertia-less motion. Far from the assemblage at r ∼ ντ there is an outer region in
which the flow field is time-dependent and the transient inertia term is comparable to
the viscous term. Note that this region is closer to the assemblage than the classical
Oseen region in which r ∼ ν/
u.
The first inertia effect scales as a 2 /ντ and it has the form of a Basset history
integral which depends on the rate of change of the net Stokes drag force exerted on
the particles in the swarm by the fluid. Performing a consistent approach which takes
into account the inter-particle interactions and considers the proper conditions on the
instantaneous boundaries of all particles, it is found that the inertia effects√
associated
with the unsteadiness of the flow contribute to particle velocities at O( Sl Re) as
long as the disturbance velocity field possesses a time-dependent Stokeslet in the far
flow field. It should be noted that this leading effect is identical for all particles in
the assemblage, since it originates √
in the outer region. Therefore, although the effect
Sl Re) the leading effect on the relative spacing
on the particle velocities
is
of
O(
√
is restricted to only o( Sl Re). Similar solution structure and memory effects due
to mutual hydrodynamic interactions were found recently by Pieńkowska (2001),
for the time intervals during which no significant evolution of the mutual positions
of particles takes place. The asymptotic analysis presented here is applicable to
larger timescales and to systems with changing geometry. On the other hand, the
theory of Pieńkowska (2001) also covers small timescales (initial transient period and
high-frequency oscillations of the external forcing) that are not described by our
approach.
Our analysis shows that, in case of low-frequency uniform oscillatory flow around
a compact assemblage of particles, the stationary history effects are independent
of the assemblage configuration and inter-particle interactions and the assemblage
response is analogous to that found for a single particle of arbitrary shape (Lovalenti
& Brady 1993). However, when the total Stokes drag force exerted on particles by
fluid √
is constant, the effect of fluid inertia on the particle velocities and spacing is
of o( Sl Re). It is anticipated that the fluid inertia associated with the convective
terms in the Navier–Stokes equations is likely to affect the particle velocities and
spacing at O(Sl Re) as in the classical case of a single particle. For the important and
most common case of a gravitational sedimentation of particle assemblage, the action
of the Basset force is limited to an initial short-lived transient period on a viscous
timescale associated with the rapid acceleration from rest. Thus, an a priori retaining
of the Basset integral in the dynamic long-time force balance as in Leichtberg et al.
(1976) can lead to erroneous results. Moreover, in this case, the theory developed here
suggests that the first-order effect of the fluid inertia is manifested at O(Re)† and it
† When there is no imposed flow and varying body force, an appropriate choice of the typical
timescale will be the Stokes timescale, τ = tS , which leads to Sl = 1.
The leading effect of fluid inertia on the motion of rigid particles
247
has contributions from both the outer and the inner regions. The convective Oseen
terms within ε2 f (V 0 , V ∞ ) in (31), are of the same order of magnitude (O(Re)) as
the unsteady terms, associated at this order with the rate of change of the stresslet
within a disturbance flow. In the inner region the inertia associated with the Eulerian
acceleration terms and with the convective terms would again affect the motion of
the particles at O(Re). These findings indicate that considering unsteady inertia terms
alone, while neglecting the convective terms, as in Feng & Joseph (1995) is not
sufficient to capture correctly the deviation from the quasi-steady theory in the case
of sedimenting particles. The entire fluid inertia should be taken into account in the
equations of motion. The leading-order effect of fluid inertia on the settling of particles
in a viscous fluid is investigated in a separate paper by Leshansky, Lavrenteva & Nir
(2003).
Another difficulty may arise when studying more general spatially inhomogeneous
undisturbed flows, like linear or parabolic flows. These flow fields are particularly
interesting from practical and experimental points of view. For these cases, the effect
of inertia within the undisturbed flow may become important at the leading order
and cannot be neglected as in the case of uniform flow. In the problem studied here
the inertia of the incident flow did not enter at the leading order since V ∞ is bounded
everywhere and thus, ε| f (V 0 , V ∞ )| |∂ V 0 /∂t| in (31). The motion of a particle in
a steady linear flow
√ was studied by Saffman (1965) who found that it experiences
a lift force of O( Re) perpendicular to the flow direction. For the slowly varying
linear flow, v ∞ = A(t) · x, the unsteady outer problem at the leading order should be
considered (the problem formulated in this paper would be modified by adding to the
left-hand side of (33) two Oseen terms: V 0 · ∇ξ V ∞ and V ∞ · ∇ξ V 0 ). The unsteadiness
of the outer field may also result from the temporal change of the inner domain in
the many-body problem even when the imposed flow is independent of time. It may
lead to the modification of the inertial lift force found by Saffman (1965) for a single
particle.
For a parabolic flow, v ∞ = B(t) : x x + . . . , it is readily seen that the Oseen terms
in the outer region dominate over the unsteady inertia terms and this may lead to
a completely different scaling for ε(Re). For example, if the particles are neutrally
buoyant, the leading term in the far disturbance field is due to a force dipole and,
therefore, f0 = ε2 in (32) and the Oseen terms balance the viscous terms in (31) for
ε = Re1/3 . This is valid only in the case of an unbounded parabolic flow. For the
Pouseuille flow in a pipe, the linear and quadratic terms in the outer flow
are of
the same order of magnitude and the expansion parameter is again ε = Rep , as
was shown by Schonberg & Hinch (1989) for small values of Rep associated with a
particle size and finite Reynolds number associated with pipe flow.
Thus, our asymptotic analysis clearly indicates that the advective terms associated
with the non-uniform incident flow cannot be neglected a priori even in the case of
the motion of the single particle studied by Maxey & Riley (1983). Therefore, the
expression for the force exerted on a particle in a non-uniform flow (Maxey & Riley
1983) should be modified by adding a contribution from the undisturbed flow inertia,
neglected in their analysis. The question concerning the first effect of inertia on the
dynamics of an assemblage embedded in spatially non-uniform flows is yet to be
answered.
This work was supported by the fund for the promotion of research at the
Technion – IIT. O.M.L. acknowledges the support of the Israel Ministry for Immigrant
Absorption.
248
A. M. Leshansky, O. M. Lavrenteva and A. Nir
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